What is Log: Definition and 632 Discussions

In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Monomials – relationships of the form



y
=
a

x

k




{\displaystyle y=ax^{k}}
– appear as straight lines in a log–log graph, with the power term corresponding to the slope, and the constant term corresponding to the intercept of the line. Thus these graphs are very useful for recognizing these relationships and estimating parameters. Any base can be used for the logarithm, though most commonly base 10 (common logs) are used.

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  1. D

    Exploring the Relationship between -ln(-∞) and ln(∞)

    For instance, say I have -ln(-∞) Does the negative sign on the natural log cancel with the negative sign on the infinity? Is this true? -ln(-∞) = ln(∞) Thank you -Drc
  2. F

    Derivative of Log Determinant of a Matrix w.r.t a scalar parameter

    Hi All, I'm trying to solve the following derivative with respect to the scalar parameter \sigma $$\frac{\partial}{\partial \sigma} \ln|\Sigma|,$$ where \Sigma = (\sigma^2 \Lambda_K) and \Lambda_K is the following symmetric tridiagonal K \times K matrix $$ \Lambda_{K} = \left(...
  3. J

    Natural Log on Radioactive Decay Formula

    Homework Statement The initial amount of radioactive atoms on a sample of 24Na is 10^10. It's half-life corresponds to 15 hours. Give the amount of 24Na atoms that will disintegrate in 1 day.Homework Equations I started to solve it using the formula N=Initial Amount of Atoms /...
  4. Q

    Why do we use the natural log in the derivative of an exponential function?

    I recently struck a question that I have not been able to find an answer to. I feel like I'm missing something obvious, so I've come here for help. The derivative of a^{x} is a^{x}lna. The explanation that Stewart 5e gives is: \frac{d}{dx}a^{x} = \frac{d}{dx}e^{(lna)x} =...
  5. H

    Natural Log : seems as a discontinous function

    The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through the domain of f exists and is equal to f(c). In mathematical notation, this is written as lim_{x\rightarrow c} f(x) = f(c) from the positive and negative sides . For ln(x) (the natural log...
  6. C

    MHB What are the solutions for t=1 in the given log equation?

    Hi guys, please help me solving this =/ \[\sqrt{\log_{a}\sqrt[4]{ax}+\log_{x}\sqrt[4]{ax}}+\sqrt{\log_{a}\sqrt[4]{\frac{x}{a}}+\log_{x}\sqrt[4]{\frac{a}{x}}}=a\]
  7. S

    Natural log (ln x) derivative question

    Homework Statement This is to help out a 40something calc student -- thank you all in advance for your help Homework Equations If f (y) = ln ln ln x, what is ∂y/∂x? The Attempt at a Solution I came up with 1/x, which I got by applying ∂y/∂x ln x = 1/x three times, is this...
  8. karush

    MHB How can we avoid making simple typos in math problems?

    Compute $\displaystyle\int_0^\frac{1}{2}\frac{4}{1-4t^2}dt$ Thot I could solve by using the log rule for integration namely: $\displaystyle\int\frac{u'}{u} = ln|u|+C$ for such $u=1-4t^2$ and $u'=-8t$ but then $\displaystyle\frac{1}{2}\int_0^\frac{1}{2}\frac{-8t}{1-4t^2}dt$ isn't going...
  9. C

    [Stat Mech] Different answers when using log space

    Homework Statement I have come across a rather interesting conundrum. Given the configurational potential energy partition function for a non-ideal gas: Z = Z_{internal}\frac{1}{N!}\left(\frac{2\pi m}{h^{2}\beta}\right)^{\frac{3N}{2}}(V^{N} - B_{2}(T)N^{2}V^{N-1}) where B_{2}(T) is the...
  10. S

    Solve Log Equations with Log Laws: Examples on Logbase 4, 2, and 10

    Homework Statement Solve for x logbase 4 for question 1. 1. log (x+2) - log (x-4) =1/2 log base 2: 2. log (3x+1) = 2 - log (x-1) log base 10: 3. log sq root (x^2 -21x) =1 similar just different numbers.. log (x-4) - log (x+2) =4 base 2 log (x-4)=4-log(x+2)...
  11. G

    True Stress-Strain Curve on the Log Scale: Which portion is the plastic region?

    My math is a little weak, so I'm having a hard time finding the elastic and plastic regions on this curve. Any further help will also be appreciated!
  12. L

    (log√125 + log √27 - √8)/ log 15 - log 2 = 3/2

    1)Without using tables, show that (log√125 + log √27 - √8)/ log 15 - log 2 = 3/2 What i tried was (3/2 log 5 + 3/2 log 3 - 3/2 log 2)/ log 5+log 3 - log2 then from here I don't know where to take it. 2) Find the value of x if log x2/ log a^2 = log y^4/logy I tried this...
  13. DeusAbscondus

    MHB Natural log and trig function question

    when I differentiate: $$ln4x+sin(x)$$ I get: $$\frac{1}{x}+cos(x)$$ and Wolfgram agrees But then when i test this by calculating indefinite integral, I get: $$ln(x)+cos(x)$$ Which leaves me with three questions: 1. what happened to the 4? 2. why isn't it integrating back to (at least)...
  14. L

    Logarithmic Expressions Simplified

    Everything is base 10 log(ab)-2logb -1 a+b/ b2 = a/b/10 = 10a/b? But in the back of the book the answer is 10b/a...
  15. L

    Given that log 2 3 = 1.585, find without using tables the

    Given that log2 3 = 1.585, find without using tables the Given that log2 3 = 1.585, find without using tables the value oflog2 ( sin ∏/3) I wrote this log2 (0.866) but was not getting the answer in the book.
  16. C

    Log transform on cylindrical coordinates

    Homework Statement I'd like to do a log transform on the radius variable of the heat conservation equation: qr - qr + Δr= ΔE/Δt where qr= -kA(dT/dr) My solution for this equation in cylindrical coordinates is: Tt+Δt=Tt+(Δt*k)/(ρ*c*Δr^2)* [(Tt-1-Tt)/(ln(rt/rt-1) - (Tt-Tt+1)/(ln(rt+1/rt)]...
  17. vrmuth

    What's the value of log 1 to the base 1 ? is it defined or not ?

    let log_{x}y ,as a function of two variable, be defined from R^{+}_{2} to R then is it continuous at (1,1) ? if so what's the image ? here the domain is D={ (x,y)/ x ε R^{+} , y\inR^{+} }. like detailed discussion :).
  18. Y

    Natural log of a sum? (not sum of natural logs)

    Homework Statement Find the derivative of y = x^2 + x^(2x)The Attempt at a Solution By looking at the equation I think I need to use implicit differentiation + natural logs. But I can't do anything with: lny = ln(x^2 + x^(2x)) So I assume I'm wrong.. Any help??
  19. N

    Integration of the natural log

    1. how to integrate∫ (ln x)2 /x dx Homework Equations The Attempt at a Solution
  20. J

    Log of 2x2 Matrix: Solving e^A= Ʃ (A^k)/k!

    Homework Statement log of 2x2 matrix [4 3] [3 4] Homework Equations e^X= A X= log A The Attempt at a Solution e^A= Ʃ (A^k)/k! then what to do?
  21. N

    Log doesn't work properly in MMA

    In[48]:= 2 Log[3] === Log[3^2] Out[48]= False Why? Seems basic enough, it should find it. In[49]:= Simplify[2 Log[3]] Out[49]= Log[9] He gets it with simplify... What am I missing this time? Thank you in advance.
  22. G

    Please correct my proof for the derivative of log x

    I found this proof on a website for the derivative of ln x: y = ln x. e^y = x, dy/dx * e^y = 1, dy/dx * x = 1, dy/dx = 1/x. My question is, why can't we use a similar method to prove the derivative of log(b)x = 1/x, like this: y = log(b) x. b^y = x, dy/dx * b^y = 1, dy/dx * x =...
  23. U

    MHB Comparing R^2 from log and level models

    Say we have 2 models: ln(y) = \beta_0 + \beta_1 x_1 + \cdots + \beta_nx_n with a known R^2 and y = \beta_0 + \beta_1 x_1 + \cdots + \beta_nx_n with a known R^2 Now I know that we can not compare the R^2's from these 2 models to determine goodness-of-fit and I am also aware of how we can...
  24. J

    Understanding the log() function

    Homework Statement Working on a computer program that will create an amortization table (a table that lists each payment on an amortizing loan). I am going to use a simple c++ array to store each row of data, and need to know the number of months required to pay off a given loan so that i...
  25. F

    How do I evaluate this log function?

    Homework Statement Homework Equations The Attempt at a Solution I assume I can't use a calculator obviously.. so I'm quite stuck. The answer is 5, but I have no idea how to get that. (log23)(log34)(log45) ... (log3132)
  26. Y

    Approximating the Sum of Log Squared Terms for Large n?

    Hi all, I have a summation series which goes like this, S = [log(1)]^2 + [log(2)]^2 + [log(3)]^2 + ... + [log(n)]^2 Since each term is square of the logarithm of a value, I think there are no general tricks which can be used to solve this. But is there any way to approximate the...
  27. C

    What is the significance of taking derivatives in the form d ln f(x) / d ln x?

    Homework Statement What does it mean when the derivative of a function f(x) is in the form: d ln f(x) / d ln x ? Is it the logarithmic scale derivative, or something? Homework Equations d ln f(x) / d ln x The Attempt at a Solution Googling.
  28. S

    Solving Log Equation Double Integration - e^(e+1)

    Homework Statement Do anyone know how to solve: y=LN x y=e+1-x Homework Equations y=LN x y=e+1-x The Attempt at a Solution LN x=e+1-x x=e^e * e^1 / e^x x*e^x=e^(e+1) then I don't know how to solve, must I solve it graphically? Actually, the original question is...
  29. L

    Integral with natural log problem

    Hello, The problem is ∫(ln x)/(x + x ln x) dx. I've done most other problems in the set, but don't know where to start with this one. Although we are just learning integration by parts, I'm not sure how this would apply. I can get to ∫u/(1+u) du Thanks for any help. Homework Statement...
  30. P

    Evaluating a log by trial and error

    Hi, I'm evaluating a log, and I'm wondering how many words I can use. When I'm trying different exponents to narrow in on the exponent that I'm looking for, can I use words next to each equation? For example, "this is not big enough" next to the numbers that aren't close to the number I'm...
  31. P

    Can I use brackets on the subscript of a log?

    Hi, can I type brackets around the subscript of a log? Can I type brackets around the non subscript as well? I included what I mean in the picture below, as it's maybe easier to see what I mean. I'm just concerned that the meaning of what I wrote changes when I include the brackets. I...
  32. baby_1

    What is the Log of a Complex Variable?

    what is log of z=X+JY? Thanks
  33. E

    For Log function, is this true?

    Hi, I have the numbers x1, ..., xN, and I need to find the max of log2(1+xi). Is this equivalent to find Log2(1+max xi), since the Log function is monotonically increasing function of its argument? Thanks
  34. F

    Converting Errors for Logarithmic Graphs for Scientists

    I think this is probably the right place to put it, didn't really fit anywhere else. I've got the joyless task of writing a lab report at university and need to put a graph in it, ideally with error bars. I don't have a problem with error bars normally, but I'm coming unstuck about what to do...
  35. M

    Troubleshooting Rolling Log Acceleration on a Ramp

    Homework Statement A uniform solid cylindrical log begins rolling without slipping down a ramp that rises 28.0 above the horizontal. After it has rolled 4.20 m along the ramp, the magnitude of its linear acceleration is closest to Homework Equations The Attempt at a Solution...
  36. E

    Solving a Logarithmic Equation with an Irrational Result

    Homework Statement 27x+1=32x+1The Attempt at a Solution log(27x+1)=log(32x+1) (x+1)log(27)=(2x+1)log(3) 2x+1=\frac{(x+1)log(27)}{log(3)} \frac{2x+1}{x+1}=\frac{log(27)}{log(3)} x=-2 I'm wondering how I would have continued solving this for the exact answer if \frac{log(27)}{log(3)} had...
  37. L

    How do you find the cubic root of n without using log keys?

    what is the quickest way to find \sqrt[3]{n} [on a pocket calculator] whitout using any \sqrt{} or log key?
  38. polydigm

    Convergence of complex log series on the boundary

    The radius of convergence of \sum\limits_{k=1}^\infty\displaystyle\frac{z^n}{n} is 1. It converges on all of the boundary \partial B(0,1) except at z=1. One way of looking at this is to analyse \sum\limits_{k=1}^\infty\displaystyle\frac{\cos n\theta}{n}+\frac{\sin n\theta}{n}. You can see the...
  39. C

    MHB Higher Prelim Exam: Solving Log Questions

    A Question from Em on Yahoo answers: Please can someone help with this question from a Higher Prelim paper? [A knowledge of how to change log bases is not a requirement of the syllabus.]7(a) Given that log_4(x)=P, show that log_16(x) =1/2P (b) Solve log_3(x)+log_9(x)=12
  40. MacLaddy

    Natural Log in a Derivative: What Are the Rules?

    Homework Statement I have a problem that I'm working on that I have almost solved, yet I am just a tad off of what the book says the answer is. I will show the way I'm doing it, and where I depart from the steps the book takes. The graph of y=(x^2)^x has two horizontal tangent lines. Find...
  41. M

    MHB Derivative with the Natural Log

    So I've come across a derivative problem that I need to solve that is showing me some of my weaknesses in my understanding/solving of Ln and e. This is what I've done so far. \(Q = 350\frac{1}{2}^{(\frac{t}{13.1})}\) \(Q = 350 * \frac{1}{2}^{(\frac{t}{13.1})}\) \(ln{Q} = \ln{350} *...
  42. L

    Derivative of s(t) = (976(.835)^t -1) + 176t using log differentiation

    1. Homework Statement The derivative of s(t) = (976(.835)^t - 1) +176t I have to take ln of both sides to bring the t down from the exponent. But I never had to apply ln to an equation of this complexity. Here is my attempt, but it doesn't even look close to being on the correct...
  43. L

    Derivative of log proof question

    Proof d/dx [log_{b}x] = d/dx (lnx/lnb) = 1/lnb d/dx(lnx) = 1/xlnb , when x>0 I think I see the constant multiple rule at work here, but why? Is it because 1/lnb is a constant, so it is "factored" out of lnx/lnb and treated as a constant while lnx is treated as the term to be...
  44. anemone

    MHB How Can I Find the Limit of the Natural Log Function in the Form of a Question?

    Find the $\displaystyle\lim_{n\to\infty}\frac{2ln(2)+3ln(3)+...+nln(n)}{n^2ln(n)}$
  45. H

    Finding the derivative, quotient rule with natural log function.

    Homework Statement Find y' of y= 1-3ln(7x)/x^4 Homework Equations The Attempt at a Solution I used the quotient rule and got: y'=x^4*d/dx(1-3ln(7x)-(1-3ln(7x)*d/dx(x^4)/(x^4)2 which is: x^4*(0-3*1/7x*7)-(1-3ln(7x))*4x^3/x^8 simplified to: 3x^4/x-1+3ln(7x)*4x^3 3x^3-4x^3+12x^3ln(7x)/x^8 take...
  46. A

    I can't find the log of a negative number

    Homework Statement there is an equation: 2^x=-6 Homework Equations The Attempt at a Solution I know that log need to be used to find x. but I can't find the log of a negative number,,
  47. P

    Understanding the Limit of Natural Log: ln(x^2-16) as x Approaches 4+

    Determine the infinite limit. lim x->4+ ln(x^2-16) I know from graphing the equation and doing a table that the limit is -infinity, but my book is saying to do the following. Let t = x^2-16, Then as x->4+, t->0+, and lim x->4+ ln(x^2-16)=lim t->0+ ln(t) by 3
  48. B

    Branch cut of the principle value log

    Homework Statement Find the branch points of g(z) = log(z(z+1)/(z-1)) and defining a branch of g as the principle branch of the logarithm find the location of the branch cuts. Homework Equations The Attempt at a Solution Since g(z) = log(z) + log(z+1) - log(z-1) the branch points are 0, 1...
  49. L

    How Do You Solve an Equation with Log for an Unknown Constant Plus Unknown?

    Homework Statement Can you solve an unknown equal to some constant+log of unknown. please see attached. All the other variables are known, except Ic1. How to you solve this? Homework Equations The Attempt at a Solution I have no idea.
  50. N

    How Large Can N Be for F(n) = Log N to Run in 1 Second?

    Hi, I'm doing this algorithm questions and i need to find the largest size n of a problem that can be solved in time t, assuming that the algorithm to solve the problem takes f(n) microseconds. For example: f(n) = log n t = 1 seconds how do i get the largest size of n in t time...
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